Basic Math Examples

$\frac{4 (\sqrt{5} + \sqrt{7})}{\sqrt{35} + 5 \sqrt{5} + \sqrt{7}}$

Step 1

Multiply $\frac{4 (\sqrt{5} + \sqrt{7})}{\sqrt{35} + 5 \sqrt{5} + \sqrt{7}}$ by $\frac{\sqrt{35} + 5 \sqrt{5} - \sqrt{7}}{\sqrt{35} + 5 \sqrt{5} - \sqrt{7}}$ .

$\frac{4 (\sqrt{5} + \sqrt{7})}{\sqrt{35} + 5 \sqrt{5} + \sqrt{7}} \cdot \frac{\sqrt{35} + 5 \sqrt{5} - \sqrt{7}}{\sqrt{35} + 5 \sqrt{5} - \sqrt{7}}$

Step 2

Multiply $\frac{4 (\sqrt{5} + \sqrt{7})}{\sqrt{35} + 5 \sqrt{5} + \sqrt{7}}$ by $\frac{\sqrt{35} + 5 \sqrt{5} - \sqrt{7}}{\sqrt{35} + 5 \sqrt{5} - \sqrt{7}}$ .

$\frac{4 (\sqrt{5} + \sqrt{7}) (\sqrt{35} + 5 \sqrt{5} - \sqrt{7})}{(\sqrt{35} + 5 \sqrt{5} + \sqrt{7}) (\sqrt{35} + 5 \sqrt{5} - \sqrt{7})}$

Step 3

Expand the denominator using the FOIL method.

$\frac{4 (\sqrt{5} + \sqrt{7}) (\sqrt{35} + 5 \sqrt{5} - \sqrt{7})}{{\sqrt{35}}^{2} + 5 \sqrt{175} - \sqrt{245} + 5 \sqrt{175} + 25 {\sqrt{5}}^{2} - 5 \sqrt{35} + \sqrt{245} + 5 \sqrt{35} - {\sqrt{7}}^{2}}$

Step 4

Simplify.

$\frac{4 (\sqrt{5} + \sqrt{7}) (\sqrt{35} + 5 \sqrt{5} - \sqrt{7})}{153 + 50 \sqrt{7}}$

Step 5

Multiply $\frac{4 (\sqrt{5} + \sqrt{7}) (\sqrt{35} + 5 \sqrt{5} - \sqrt{7})}{153 + 50 \sqrt{7}}$ by $\frac{153 - 50 \sqrt{7}}{153 - 50 \sqrt{7}}$ .

$\frac{4 (\sqrt{5} + \sqrt{7}) (\sqrt{35} + 5 \sqrt{5} - \sqrt{7})}{153 + 50 \sqrt{7}} \cdot \frac{153 - 50 \sqrt{7}}{153 - 50 \sqrt{7}}$

Step 6

Combine fractions.

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Step 6.1

Multiply $\frac{4 (\sqrt{5} + \sqrt{7}) (\sqrt{35} + 5 \sqrt{5} - \sqrt{7})}{153 + 50 \sqrt{7}}$ by $\frac{153 - 50 \sqrt{7}}{153 - 50 \sqrt{7}}$ .

$\frac{4 (\sqrt{5} + \sqrt{7}) (\sqrt{35} + 5 \sqrt{5} - \sqrt{7}) (153 - 50 \sqrt{7})}{(153 + 50 \sqrt{7}) (153 - 50 \sqrt{7})}$

Step 6.2

Expand the denominator using the FOIL method.

$\frac{4 (\sqrt{5} + \sqrt{7}) (\sqrt{35} + 5 \sqrt{5} - \sqrt{7}) (153 - 50 \sqrt{7})}{23409 - 7650 \sqrt{7} + 7650 \sqrt{7} - 2500 {\sqrt{7}}^{2}}$

Step 6.3

Simplify.

$\frac{4 (\sqrt{5} + \sqrt{7}) (\sqrt{35} + 5 \sqrt{5} - \sqrt{7}) (153 - 50 \sqrt{7})}{5909}$

Step 6.4

Group $153 - 50 \sqrt{7}$ and $\sqrt{5} + \sqrt{7}$ together.

$\frac{4 ((153 - 50 \sqrt{7}) (\sqrt{5} + \sqrt{7})) (\sqrt{35} + 5 \sqrt{5} - \sqrt{7})}{5909}$

Step 7

Expand $(153 - 50 \sqrt{7}) (\sqrt{5} + \sqrt{7})$ using the FOIL Method.

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Step 7.1

Apply the distributive property.

$\frac{4 (153 (\sqrt{5} + \sqrt{7}) - 50 \sqrt{7} (\sqrt{5} + \sqrt{7})) (\sqrt{35} + 5 \sqrt{5} - \sqrt{7})}{5909}$

Step 7.2

Apply the distributive property.

$\frac{4 (153 \sqrt{5} + 153 \sqrt{7} - 50 \sqrt{7} (\sqrt{5} + \sqrt{7})) (\sqrt{35} + 5 \sqrt{5} - \sqrt{7})}{5909}$

Step 7.3

Apply the distributive property.

$\frac{4 (153 \sqrt{5} + 153 \sqrt{7} - 50 \sqrt{7} \sqrt{5} - 50 \sqrt{7} \sqrt{7}) (\sqrt{35} + 5 \sqrt{5} - \sqrt{7})}{5909}$

Step 8

Simplify each term.

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Step 8.1

Multiply $- 50 \sqrt{7} \sqrt{5}$ .

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Step 8.1.1

Combine using the product rule for radicals.

$\frac{4 (153 \sqrt{5} + 153 \sqrt{7} - 50 \sqrt{5 \cdot 7} - 50 \sqrt{7} \sqrt{7}) (\sqrt{35} + 5 \sqrt{5} - \sqrt{7})}{5909}$

Step 8.1.2

Multiply $5$ by $7$ .

$\frac{4 (153 \sqrt{5} + 153 \sqrt{7} - 50 \sqrt{35} - 50 \sqrt{7} \sqrt{7}) (\sqrt{35} + 5 \sqrt{5} - \sqrt{7})}{5909}$

Step 8.2

Multiply $- 50 \sqrt{7} \sqrt{7}$ .

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Step 8.2.1

Raise $\sqrt{7}$ to the power of $1$ .

$\frac{4 (153 \sqrt{5} + 153 \sqrt{7} - 50 \sqrt{35} - 50 ({\sqrt{7}}^{1} \sqrt{7})) (\sqrt{35} + 5 \sqrt{5} - \sqrt{7})}{5909}$

Step 8.2.2

Raise $\sqrt{7}$ to the power of $1$ .

$\frac{4 (153 \sqrt{5} + 153 \sqrt{7} - 50 \sqrt{35} - 50 ({\sqrt{7}}^{1} {\sqrt{7}}^{1})) (\sqrt{35} + 5 \sqrt{5} - \sqrt{7})}{5909}$

Step 8.2.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{4 (153 \sqrt{5} + 153 \sqrt{7} - 50 \sqrt{35} - 50 {\sqrt{7}}^{1 + 1}) (\sqrt{35} + 5 \sqrt{5} - \sqrt{7})}{5909}$

Step 8.2.4

Add $1$ and $1$ .

$\frac{4 (153 \sqrt{5} + 153 \sqrt{7} - 50 \sqrt{35} - 50 {\sqrt{7}}^{2}) (\sqrt{35} + 5 \sqrt{5} - \sqrt{7})}{5909}$

Step 8.3

Rewrite ${\sqrt{7}}^{2}$ as $7$ .

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Step 8.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{7}$ as $7^{\frac{1}{2}}$ .

$\frac{4 (153 \sqrt{5} + 153 \sqrt{7} - 50 \sqrt{35} - 50 {(7^{\frac{1}{2}})}^{2}) (\sqrt{35} + 5 \sqrt{5} - \sqrt{7})}{5909}$

Step 8.3.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{4 (153 \sqrt{5} + 153 \sqrt{7} - 50 \sqrt{35} - 50 \cdot 7^{\frac{1}{2} \cdot 2}) (\sqrt{35} + 5 \sqrt{5} - \sqrt{7})}{5909}$

Step 8.3.3

Combine $\frac{1}{2}$ and $2$ .

$\frac{4 (153 \sqrt{5} + 153 \sqrt{7} - 50 \sqrt{35} - 50 \cdot 7^{\frac{2}{2}}) (\sqrt{35} + 5 \sqrt{5} - \sqrt{7})}{5909}$

Step 8.3.4

Cancel the common factor of $2$ .

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Step 8.3.4.1

Cancel the common factor.

$\frac{4 (153 \sqrt{5} + 153 \sqrt{7} - 50 \sqrt{35} - 50 \cdot 7^{\frac{2}{2}}) (\sqrt{35} + 5 \sqrt{5} - \sqrt{7})}{5909}$

Step 8.3.4.2

Rewrite the expression.

$\frac{4 (153 \sqrt{5} + 153 \sqrt{7} - 50 \sqrt{35} - 50 \cdot 7^{1}) (\sqrt{35} + 5 \sqrt{5} - \sqrt{7})}{5909}$

Step 8.3.5

Evaluate the exponent.

$\frac{4 (153 \sqrt{5} + 153 \sqrt{7} - 50 \sqrt{35} - 50 \cdot 7) (\sqrt{35} + 5 \sqrt{5} - \sqrt{7})}{5909}$

Step 8.4

Multiply $- 50$ by $7$ .

$\frac{4 (153 \sqrt{5} + 153 \sqrt{7} - 50 \sqrt{35} - 350) (\sqrt{35} + 5 \sqrt{5} - \sqrt{7})}{5909}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$\frac{4 (153 \sqrt{5} + 153 \sqrt{7} - 50 \sqrt{35} - 350) (\sqrt{35} + 5 \sqrt{5} - \sqrt{7})}{5909}$

Decimal Form:

$0.98911498 \dots$